Cutscenes Blender Game Engine (BGE)
What you'd basically do is create a scene with a plane and a camera pointed at the plane, and make the plane use your cutscene or video file as its texture. Then, using a Delay logic brick or something similar, you'd switch from that scene to another scene, which contains the game stage or level. 

If you want a live animation to play, just animate the camera and the objects in the first scene, and use a Delay sensor again. 

This video does a good job of explaining a method of implementing the same method: https://www.youtube.com/watch?v=P2W_fQayYcA It's a video by SuperGloop.